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In this paper doubly
transitive and 3/2-transitive permutation groups are classified under hypotheses
somewhat weaker than solvability. We mention two examples.
Let 𝒮(pn) denote the group of semilinear transformations over GF(pn). The
following combines a result of Huppert on solvable 2-transitive groups and a result of
Zassenhaus on sharply 2-transitive groups.
Theorem I. Let G be a p-solvable doubly transitive permutation group with
Op(G)≠⟨1⟩. Then deg G = pn for some n and we have one of the following: (i)
G ⊆𝒮(pn), (ii) G is solvable and pn = 32,52,72,112,232 or 34, or (iii) G is
nonsolvable and pn = 112,192,292 or 592.
The second result reads better as a theorem on linear groups.
Theorem II. Let group G act faithfully on vector space V over GF(p) and let G
act 1/2-transitively but not semiregularly on V#. If G is imprimitive as a linear
group, then G is solvable and we have one of the following: (i) |V| = p2n for p≠2 and
G is a specific group of order 4(pn − 1), (ii) |V| = 34 and |G| = 32, or (iii) |V| = 26
and |G| = 18.
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